CC BY
11
УДК 517.552, 515.165
The Hodge Filtration on Complements of Complex Subspace Arrangements and Integral Representations of Holomorphic Functions
Yury V. Eliyashev*
Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 10.01.2013, received in revised form 18.02.2013, accepted 18.03.2013 We compute the Hodge filtration on cohomology groups of complements of complex subspace arrangements. By means of this result we construct integral representations of holomorphic functions such that kernels of these representations have singularities on subspace arrangements.
Keywords: Hodge filtration, plane arrangements, integral representations, toric topology.
Introduction
A study of topology of coordinate subspace arrangements appears in different areas of mathematics: in toric topology and combinatorial topology [3,4], in the theory of toric varieties, where complements to coordinate subspace arrangements play the role of homogeneous coordinate spaces [5,6], in the theory of integral representations of holomorphic functions in several complex variables, where coordinate subspace arrangements play the role of singular sets of integral representations kernels [1,10].
The universal combinatorial method for the computation of cohomology groups of complements to arbitrary subspace arrangements was developed in the book of Goresky and Macpher-son [8] (see also [11]), but this method often leads to cumbersome computations. In the study of toric topology, in particular, in works of Buchstaber and Panov [3,4], the method for the computation of the cohomology of complements to coordinate subspace arrangements was developed, this method is simpler than the universal method and allows to get some additional topological information.
The main purpose of this article is to compute the Hodge filtration on the cohomology rings of complements to complex coordinate subspace arrangements. We will show that the Hodge filtration is described by means of a special bigrading on the cohomology rings of complements to complex coordinate subspace arrangements, which was introduced in [3,4], this bigrading was obtained originally from the combinatorial and topological ideas. We use these results to construct the integral representations of holomorphic functions such that the kernels of these representations have singularities on coordinate subspace arrangements.
The first section of this paper consists of different facts about topology of complements to complex coordinate subspace arrangements, in the text of this section we follow [3,4]. Let Z be a complex coordinate subspace arrangement in Cn. In [3,4], from the topological reasons, the differential bigraded algebra R was introduced (R is determined by combinatorics of Z) such that the ring of cohomology H*(Cn \ Z) is isomorphic to the ring of cohomology H*(R). Denote
* eliyashev@ mail. ru © Siberian Federal University. All rights reserved
by Hp'q (R) the bigraded cohomology of the algebra R, then
Hs(C" \ Z) ~ 0 Hp'q (R).
p+q=s
Thus, we get a bigrading on the cohomology ring H* (Cn \ Z).
In the second section we recall some facts and concept from differential topology and complex analysis. These facts we use in the last two sections.
In the third section the main theorem of this paper is proved. We will show that the bigrading on the cohomology of R and, consequently, the bigrading on the cohomology H* (Cn \ Z) appear naturally from the complex structure on the manifold C" \ Z. In particular, denote by Fk Hs(C" \ Z, C) a k-th term of the Hidge filtration on Hs(C" \ Z, C). Then there is the following theorem.
Theorem 1.
FkHs(C" \ Z, C) = 0 Hp's-p(R, C). p>k
In the last section we construct integral representations of holomorphic functions such that kernels of these representations have singularities on coordinate subspace arrangements.
1. General facts on topology of coordinate subspace arrangements
In this section different facts about topology of complements to coordinate subspaces arrangements are gathered. All statements of this section are taken from [4].
Let K be an arbitrary simplicial complex on the vertex set [n] = {1,..., n}. Define a coordinate planes arrangement
Zk := (J La,
where a = {h, ..., im} C [n] is a subset in [n] such that a does not define a simplex in K and
La = {z e C" : = • • • = zim = 0}.
Any arrangement of complex coordinate subspaces in C" of codimension greater than 1 can be defined in this way.
Consider a cover UK = {Ua}aeK of C" \ ZK, where
Ua = C" \(J{* =0}.
i0a
By Da x SY denote the following chain
D2a x Si = {|zi| < 1 : i e a; |zj| = 1 : j e 7, Zk = 1 : k g 7 U a},
where a, 7 C [n] and a n 7 = 0. We define the form
dz1 = dzil ^ ^ dzik (1) Z1 Zii Zik
where I C [n], |11 = k, I = {i1,..., ik}, and i1 < • • • < ik.
The orientation of the chain D^ x Si is such that the restriction of the form
1 dzY
(7-ï)M O; A A(V=Lfoj A ^
on D2a x Si is positive. Then the boundary of this chain equals
3Dl x S'Y = Sie<7(-l^D2x ,
where (¿,7) is the position of i in the naturally ordered set 7 U i. Definition 1. The topological space
Zk = U D x S[
x S[1n]\CT aeK
is called a moment-angle complex.
Theorem 2 ( [4]). There is a deformation retraction from Cn \ Zp to Zp.
Definition 2. A Stanley-Reisner ring of a simplicial complex K on the vertex set [n] is a ring
Z[K] = Z[vi,...,vn]/IK,
where Ik is a homogeneous ideal generated by the monomials vJ = Пi£j vi such that a £ K :
Ik = (vi! •... • Vim : {¿i, ..., ¿m} £ K).
Consider a differential bigraded algebra (R(K),Sn) :
Rk :=Л[м1,...,м„] <g> Z[K]/J,
where Л[и1;..., un] is an exterior algebra, J is the ideal generated by monomials v2, ui ® vi, i = 1,..., n. Bidegrees of generators vi, ui of this algebra are equal to
bideg vi = (1,1), bideg ui = (1,0).
The differential SR is defined on the generators as follows
SRU = vi, Snvi = 0.
Remark 1. In [4] a different bigrading on the algebra Rp was used, but our bigrading is equivalent to the bigrading from [4].
We denote by R^9 the homogeneous component of the algebra RK of the bidegree (p, q). The differential Sn is compatible with the bigrading, i.e., Sn(RK'q) Q R/cq+1. Consider the complex
Sr RP'9-1 SR RP'9 RP'9+1 SR
K K K '
denote by Hp'q (RK) a cohomology group of this complex. It is clear that the cohomology of RK are isomorphic to
Hs(RK)= 0 Hp'q(RK).
p+q=s
Theorem 3 ( [4]). The cohomology ring H* (Cn \ Zp) is isomorphic to the ring H*(Rp).
Remark 2. The relation between Theorem 3 and the results of Goresky and Macpherson [8] on cohomology of subspace arrangements is described in [4, Ch. 8].
Now we describe the explicit construction of the isomorphism of Theorem 3. First, we construct a cell decomposition of ZK. Define a cell
ECT7 = {N < 1 : i e a; |zj| = 1, zj = 1 : j e 7; zfc = 1 : k g 7 U a},
where a, 7 C [n] and a n 7 = 0. The closure of this cell equals ECT7 = D^ x S^. The orientation of ECT7 is defined by the orientation of D2 x S^. We obtain the cell decomposition
Zk = ECT7.
Let C*(ZK) be the group of cell chains of this cell decomposition, then denote by C*(ZK) the group of cell cochains. Let E^7 be a cocell dual to the cell ECT7, i.e., E^7 is a linear functional from C*(ZK) such that (E^7, ECT'y} = ¿^Z^ (the Kronecker delta).
Denote w/vj := ... <g> Vj^ ... vjp, where I = {ii,..., iq}, ii < • • • < iq, J = {ji,..., jp}, and I n J = 0,1, J C [n], (we suppose that wgvg = 1).
Proposition 1 ( [4]). The linear map ^ : Rk ^ C*(Zk), ) = E^Y is an isomorphism of
differential bigraded modules. In particular, there is an additive isomorphism H*(Rk) ^ H*(Zk).
From the structure of the cell decomposition of ZK and Theorem 2 we obtain that every cycle r e Hs(Cn \ ZK) has a representative of the form
r= E rp,q, (2)
p+q=s
where rp,q is a cycle of the form
rp,q = E • D2 X S'Y. (3)
M=9
|Y|=p-q
A group generated by all cycles of the form (3) is denoted by (Cn \ ZK). Obviously we have
Hs(Cn \ Zk)= 0 (Cn \ ZK).
p+q=s
It follows from Proposition 1 that (rp,q)} = 0 for any rp,q G (Cn \ ZK) and G (RK), p' = p and q' = q. Hence, the pairing between rp,q and ) can be
nonzero only if p' = p, q' = q. Therefore the pairing between the vector spaces (Cn \ ZK, R) and (RK ® R)) is nondegenerate if p = p', q = q' and equals to zero otherwise.
2. Cech cohomology, filtrations and cochains
In this section we recall some facts from differential topology and complex analysis, we mainly use a material from the books [2,9]. Let X be a complex manifold and U = {Ua}aeA is an open, countable, locally finite cover of this manifold. Now we introduce the following notation for sheafs on X: Es denotes the sheaf of ^^-differential forms of degree s, £p'q denotes the sheaf of C^-differential forms of bedegree (p, q), denotes the sheaf of holomorphic differential forms of degree p.
Definition 3. The decreasing filtration
Fk E • = 0 Ep'"-p,
on the de Rham complex (E*, d) is called the Hodge filtration.
The Hodge filtration induces a filtration FkHs(X, C) on a de Rham cohomology, i.e.,
FkHs(X, C) = Im(Hs(FkE*(X),d) — Hs(E*(X),d)),
where Hs(E*(X),d) is the cohomology of the de Rham complex and Hs(FkE*(X),d) is the cohomology of k-th term of the Hodge filtration. In other words, if ш lies in FkHs (X, C) then there is a form ш, [ш] = ш such that
where £ Ep'q (X).
Let C4(Es, U) be the Cech-de Rham double complex for the cover U: C4(Es, U) with a Cech coboundary operator S : C4(Es,U) — Ci+1(Es,U) and a de Rham differential d : C4(Es,U) — C4(Es+1, U) on this complex, i.e.,
t+i
(Mi0,..'i*+i = (-1)^ (-1)j 4o'...'j '...'it+1 |Ui0 n-nwit+1 ,
j=0
(d^i0'...'it = d(ш)io '...'it.
The associated single complex is defined by
Kr(U,E•)= 0 C*(Es,U)
^ tc
s+t=r
the operator D = S + d is the differential of this complex. Notice that our definition of Cech coboundary S is different from the standard one by the factor ( — 1)s, with this choice of sign we get D2 = 0, hence (K*(U,E*),D) is a complex. There is a natural inclusion of the de Rham complex £ : E*(X) — C0(E•,U), е(ш)^0 = ш|ц. , also we denote the induced map from E*(X) to K *(U, E •) by £.
Theorem 4 ( [2]). The inclusion £ : E*(X) — K*(U) is a quasi-isomorphism of complexes, i.e., Hs(X, C) ~ Hs(K*(U, E•), D).
The Hodge filtration FkK*(U,E*) is defined naturally on (K*(U,E*),D). This filtration induces a filtration on cohomology FkHs(K*(U, E*), D). There is an isomorphism FkHs(X, C) ~ F k H s(K *(U, E *), D).
Consider a subcomplex Kr (U, П*) of the complex Kr (U, E*)
Kr(U, fi*)= 0 Ct(^s,U),
^ t
s+t=r
and an inclusion map t : Kr(U, Q*) — Kr(U, E• ). It is easy to get the following statement.
Theorem 5. Suppose U is a d-acyclic cover of X then the inclusion t is a quasi-isomorphism, of the complexes K * (U, Q*) and K *(U, E * ).
Let FkQp be a stupid filtration on the de Rham complex of holomorphis forms (Q*, d), i.e.,
FkQp = i Qp for p > k, | 0 for p < k.
The stupid filtration induces filtration on cohomology FkHs(K*(U, Q*), D). Suppose U is a d-acyclic cover of X then FkHs(K*(U, Q*),D) ~ FkHs(X, C).
From now until the end of this section we will follow the paper [7].
Definition 4. A U-chain of degree t and of dimension s on the manifold X is an alternating function r from the set of indexes At+1 to the group of singular chains in X of dimension s such that r is nonzero on a finite number of points from At+1 and
supp(ri0i...jit ) C Uio n • • • l~l Wit,
for every (io,..., it) G At+1, where supp(ri0i...jit ) is the support of the chain Fj0j...jjt.
Let Ct,s(U) be an additive group of U-chains of degree t and of dimension s on the manifold X. Define maps J' : Ct,s(U) ^ Ct_1s(U)
(¿'r)io.....it-1 =(-1)S E ri.io-..it-! ,
¿eA
and d : CM(U) ^ Ct,s_1(U)
(d r)io,...,it = d(r)io.....¿t,
i.e., the operator d is a boundary operator on each chain rioj...jit. The groups Ct,s(U),t, s > 0 together with the differentials J', d form a double complex. Define a map e' : C0,s(U) ^ Cs(X) in the following way
e'(r)^Ti.
¿eA
Now we will construct a pairing between elements of Ct,s(U) and Ct(Es, U). Suppose r G Ct,s(U) and w G Ct(Es, U), then
<w,r> = E /r, , -¿o,...,it.
There are the following relations for the pairing:
<wt,s,d rt,s+1> = <dwt,s, rM+1>,
<Jwi's, rt+1,s> = <wt's,J'rt+1,s>,
f ws = <ews, To,« >, ^'(ro.s)
where wt-s G Ct(Es,U), ws G Es(X), and rt,s G Ct,s(U).
Definition 5. Let r be a singular cycle of dimension s on X, then a U-resolvent of length k of the cycle r is a collection of U-chains r® G CijS_j(U), 0 ^ i ^ k such that r = e'r0 and
dr® = -J'ri+1.
Proposition 2. Given an s-dimensional cycle r, a closed differential form w of degree s, a
U-resolvent r0,..., rk of the cycle r and a cocycle w G Ks(U) such that w = J2 w®'s-i, wi,s-i G
¿^k
C®(£s-i,U) and the cocycle ew is cohomologous to w in Hs(K*(U,E*),D), then
E<wi,s-i, r®>. i^k
This proposition follows directly from the properties of the pairing.
/r w =
3. The Hodge filtration of cohomology of complements to coordinate subspace arrangements
In this section we compute the Hodge filtration of the cohomology ring H* (Cn \ Zp, C). It
follows from Theorem 3 and Proposition 1 that there is the isomorphism H*(Cn \ Zp, C) —
H*(RK <g> C).
Theorem 1. Let Hp'q(Rp <8> C) be the bigraded cohomology group of the complex Rpq <g) C, then there is an isomorphism
FkHs(Cn \ ZK, C) - 0 Hp's-p(Rp <g> C).
Proof. First, we will prove the lemma. Lemma 1. Let
ГР>9 "У - ^ X S'
M=q M=P-q
be a cycle in Cn \ Zk. Then there is a Uk-resolvent of the cycle rp,q of length q:
r0 rq
P,q' ' ' ' ' P,q>
where rp q is a UK-chain of dimension q + p — k and of degree k of the form
(rp,q)ao,...,afc ^^ CCT7,ao...afc • D<r x S-y.
M=q-fc
|7|=P-9+k
Proof. We will use the induction on the length k of the resolvent. We going to construct the resolvent of the special form
(rp,q)CTfc,CTfc-i,...,CTo ^^ CCTfc7,CTfc...CTo • D<rfc X Si,
M=p-q+k
for |<7j | = q—j, aj C j > t j = 0,..., k (in other words, (aj} is a chain of subsets C • • • C a0, and |aj| = q — j); and (r^)ao,...,ak = 0 for any other indexes a0,..., ak.
The base of induction: define (rp )CTo = CCTo7 • D^o x S^ with |a0| = q and (rp )a = 0
|7|=p-q
for any other indexes a. We get
rp,q ^ [ CCTO7 • D2o X S7 = y y(r0.q= £ ^p.q, |ao|=q ctgK
M=p-q
therefore rp.q is the resolvent of length 0.
Suppose that the resolvent rp.q,..., rk,q of length k is already constructed. Recall that (i, 7) is the position of i in the naturally ordered set 7 U i. Define
(rp+q )CTfc\i,CTfc...CTo = ( — l)P+q ^^ ( —1)( '7)CCTfc7,CTfc...CTo D<rfc\i X S-
YUi,
|Y|=p-q
for i G , |aj | = q — j, aj+i С <7j, and
(rfc+1) =0
V p,q Mc -,afc + i
for any other indexes ao,..., ak+i. Let us show that rP _,..., rk +1 is a resolvent of length k +1 :
- +1W...q = (-1)P+q k E(rP
- p,q V / ^ V P , q
ie^fc
^^ ( —1)( ' 7)CCTfc7 , Tfc...To D<rfc\i X SiUi ^^ CCTfc7 , CTfc ...To dD(Tfc X S7 = , q )CTfc...ffo .
ie^fc |yI=P—q
|7|=p-q
For any indexes a0,..., ak different from ak,..., am+1, am—i,..., a0, 0 < m ^ k, directly from definition of rk,q, rk+q1, we get
(drp,q)ao,...,afc = —)a0,...,al =
Consider the last case ak \ i, ak,..., am+1, am—1,..., a0, for 0 < m < k. Since by the induction hypothesis rp q ... rk q is a resolvent, — JTi q = drk —q1, hence we have J'drk q = 0, and
p q p q p q p q p q
(A'drfc )
(-i)p+q-k+1 ^ EE (-i)(i ' 7)c^Y , ...ffQ ■ \i X Siui = 0.
0m.+iCo-mCa-m_i |Y|=P-q+k iGCTfc
|.m|=q—m
Therefore, for a fixed i G ak, we get
E E (-1)(i'7)CCTfc7,CTfc...CTQ ■ D2k\i X S7Ui =
^m+lC^mC^m-l |y|=^ —q+k
|.m|=q—m
On the other side,
= ( — 1)p+q— k+1 £ £ ( —1)(i,7)CTfc7,Tfc...To • D2fc\i X Siui,
Tm+lCTmCTm-l |7|=p —q+k
|Tm|=q—m
hence (i'rk+1)Tk\i,Tfc...Tm+lTm-1...To = 0. We h^e shown that = —¿Tp+1. □
It follows from Theorem 2 and the construction of the cell decomposition of the moment-angle complex ZK that any cycle rs e Hs(Cn \ ZK) can be represented as a sum of the cycles rp,q :
rs = E rp,q,
p+q=s p>q
where rp,q is the cycles of the form (3). From Lemma 1 we have the construction of the resolvent rp.q,..., rp.q of the cycle Fp,q.
The cover Uk is d-acyclic. Indeed, all elements of the cover and their intersections are isomorphic to Cn—k x (C*)k for appropriate choice of k and consequently are a Stein manifolds.
From Theorem 4 and Theorem 5 we obtain that Hs(K'(Uk,fi'),D) is isomorphic to the de Rham cohomology group Hs(Cn \ ZK, C). Recall that we use the following notation for the inclusions of complexes
e : E'(Cn \ ZK) '(uK,E•), T : K'(uK, fi') K'(uK, E•). We will use the same notation for the induced isomorphisms on the cohomology groups:
Hs(Cn \ ZK, C) ~ Hs(K'(UK, E'),D) ~ Hs(K• (uK, fi'),D).
Lemma 2. Let w G Hs(K*(Uk, ),D), then there is a cocycle w such that w = w in Hs(K'(Uk, ),D), to = £p+g=s , G Cq(Uk, 0p), and
l"l=P "
where Dwp'q = 0 for any p and q.
Proof. Consider an arbitrary element w of Hs(K*(Uk, 0*),D), this element is representable by cocycle w = £p+q=s , where wp,q G Cq(Uk, 0p) and ¿wp-q = —dwp-1-q+1. The cocycle w has a unique decomposition w = w + V, where (wp'q)CT0i...jCTq is the following form
|-|=p T"
and the Laurent expansion of (Vp'q)CT0i...jCTq does not contain summands —".
z-
Let us show that w and V are cocycles. Since w is a cocycle, we have &op'q + ¿Vp'q =
—dwp-1>q+1 — dVp-1'q+1. The forms —" are closed, hence dwp-1>q+1 = 0. Since the Laurent
z-
expansions of the components of the cochain ¿Vp'q do not contain summand-and the cochain
~ T"
&op'q can be exact if and only if &op'q = 0 (because nonzero linear combinations of the forms
—" are nonexact on any elements of the cover Uk), &op'q = 0. We get that &op'q = dwp'q = 0, z-
consequently, w is a cocycle. The cochain V = w — w is a difference of two cocycles, hence V is a cocycle.
Now we going to show that V is a coboundary.
-1
Lemma 3. Let r G Hs(C \ then /r e-1 o t= 0.
r'
Proof. For the cycle r we have the expansion to the sum r = £ rp,q. Lemma 1 gives the
p+q=s
explicit construction of the resolvent r0,q,..., rqq of the cycle rp,q. Let us construct a cocycle / = £ cohomologous to tof K*(WK, E•)
p+q=s
^ ^p'q for q<k,
V^'9 = ^ - d£-1(^p-1'q+1 - d^-1 (^p-2'q+2 - d£-1(----d£-1«'p+q)))) for q=k,
0 for q>k.
From Proposition 2 we obtain
/ £-1 ◦ t(V) = e E(rP ,q,vrk • k).
p+q=s fc^q
Let k < q and w G np+q-k(UCT'), it is easy to see, that w|D2 xSi =0 for |a| = q — k > 0, |y| = p — q + k, a C a'. The forms (V^-^)^,...,^ are holomorphic on Ua0 n • • • nUafc, indeed, (^s-fc.fc)a0 ...,afc = (Vs-fc,fc)a0i...,afc, on the other side, (rk,q)a0v..iQk is a linear combination of the chains D2 xS^, |a| = q — k > 0, |y| = p—q+k, combining these two facts we get (r^, V^-^) = 0.
Consider the case k = q. From the definition of V it follows that Vp'q = Vp'q + for some ^ G Cq(Ep-1,U). Since (rp )a0j...,ak is a linear combination of the cycles S^, |y| = p,
S1 = {|Tj | = 1: j G Y, Tk = 1: k £ 7>,
(rp,q, //>«-q'q) = (rp,q, /s-q'q)• Indeed, by the Stokes formula jSl d^ = 0, hence (rp,q, dy) = 0.
= /r? »/.«-q
" p,q> rq ) ( P;?'
Expand the from (/s-q,q)a0i...,a„ to the Laurent series,
i,/,s-9,q) = Y^ Y^ C T —
(/ )ao,...,a, = / y / y Co,i,ao...a, z1 • • • zn •
a=(ai ,...,an|1|=p
f dz
The integral z^1 • • -z^"- is nonzero only if I = 7 and a = 0, i.e., for the forms ^r31, but
by the construction of /s-q,q the Laurent expansion of (/s-q'q)ao,...,aq does not contain the dz
summands —- • Consequently, (rp,q,/s-q,q) = 0^
We have shown that /r e-1 o t(/) = 0^ Lemma 3 is proved. □
By the de Rham Theorem any closed form w of degree s on Cn \ Zk is exact if and only if /r w = 0 for any cycle r G Hs(Cn\ZK) It follows from Lemma 3 that e-1 ot(/) is cohomologous to zero, hence / is a coboundary . Lemma 2 is proved. □
By Lemma 2 any cocycle w G Hs(K*(WK, Q*), D) is cohomologous to w = $^p+q=s wp'q, where wp'q is of the form (4). Moreover, wp'q G Cq(Wk, Qp) is a cocycle, i.e., Dwp'q = 0^ We denote by Hp'q(K*(Wk, Q*), D) a subspace of H«(K*(Wk, Q*), D) generated by cocycles Wp'q• We obtain
H«(K*(WK,Q*),D)= 0 Hp'q(K*(WK,Q*),D)
p+q=s
Then the filtration FkHs(K*(WK, Q*),D) equals
FkH«(K*(WK, Q*),D) = 0 Hp'«-p(K*(WK, Q*),D)
p>k
Hence,
FkHs(Cn \ ZK, C) e ~°T 0 Hp's-p(K*(WK, Q*),D)
Pfc hs(cn \ zk C)e ~°T 0 hp,s-p ( p>k
By the same argument as in Lemma 3 we obtain that for every cycle rp,q G Hp,q (Cn \ ZK), and every cocycle wp 'q G Hp 'q (K*(WK, Q*), D), the following equality holds
i e-1 o t(wp''q') = 0,
for p = p', q = q'^ It follows from nondegeneracy of the pairing between cohomology and homology that the pairing between elements of Hp,q(Cn\ZK, C) and e-1 ot(Hp'q (K*(WK, Q*),D)) is non-degenerate if p = p', q = q' and equals to zero otherwise. Thus, e-1 ot(Hp 'q (K*(WK, Q*), D)) = ^(Hp''q'(RK ® C)) □
4. Integral representations of holomorphic functions
In the last section we study integral representations of holomorphic functions such that kernels of these integral representations have singularities on coordinate subspace arrangements in Cn. The examples of such integral representations are the multidimensional Cauchy integral representation, whose kernel has singularity on ({z1 = 0} U • • • U {zn = 0}), and the Bochner-Martinelli integral representation, whose kernel has singularity on {0}. In [10] a family of new integral representations of this kind was obtained, the kernels of these integral representations have singularities on the subspace arrangements defined by simple polytopes.
Denote by U the unit polydisc in Cn :
U = {z = (zb. ..,zn) e Cn : |z,| < 1, i = 1, ...,n}. Notice that the moment-angle complex Zk is lying on the boundary dU of the polydisc.
Theorem 6. Given a nontrivial element ш' from FnHs(Cn \ Zk, C). Then there exists a closed (n, s — n)-form ш, [ш] = ш' and an s-dimensional cycle Г in Cn \ Zk with support in Zk, such that for any function f holomorphic in some neighborhood of U the following integral representation holds
f (Z ) = j f (zMz — Z)
for z e U.
Proof. Since ш' e FnHs(Cn \ ZK, C), by Theorem 1 there is a cycle Г e Hs(Cn \ ZK, C),
Г \ Ccty • Dir X
|a| = s-n |Y|=2n-s
such that (Г, ш'} = 1. It follows from Lemma 2 that there exists a cocycle шп,г!-п e Cs-n(WK, ^n), ^n'S ")a„,...,as_„ = Bao,...,aI_„-1 Л • • • Л -П,
z1 zn
that is cohomologous to т-1 ое(ш') in Hs(K*(WK, П*), D). The form ш = е-1 о(—d6-1)s-n^n's-n is cohomologous to ш', so J ш = 1.
Let us show that ш and y define an integral representation. Consider the integral J ш^ — Z),
where Z e U, here the notation ш(z — Z) stands for the form ш after the change of coordinates z ^ z — Z. Notice that the form o>(z — Z) is closed in U, thus the integral of this form depends only
on the homological class of the integration cycle. Let us make a change of coordinates / ш^)
Jr-c
where Г—Z is a cycle Г shifted by the vector —Z. In the sequel we will use the subindex —Z to denote chains, cycles, and sets in Cn shifted by the vector —Z.
Let us show that Г — Z is homologous to Г. Notice that (ZK — Z) П ZK = 0 for any Z e U. Indeed,
Zk — Z = U (D2 X Snv — Z), ZK = U ,
we see that (D2 x — Z) П LCT' = 0 for any a e K, a' g K and Z e U. Consider the
chain Г-z = {y : y = x — tZ, x e Г, t e [0,1]}, the support of the chain is a subset of Uie[o 1] (Zk — tZ), therefore is a subset of Cn \ ZK. Its boundary equals = (Г — Z) — Г, i.e., (Г — Z) and Г are homologous. So we have returned to the case /Г ш^), which was already considered. We get
/^(z — Z )=/ сф) = 1.
Jr Jr-z
By dentition ш(z — Z) is an (n, s — n)-form. Let f (z) be a function holomorphic in some neighborhood of unit polydisc U. Since the operators д and 6 are interchangeable with the multiplication by a holomorphic function, we get f (z) • ш^ — Z) = е-1 о (—d6-1)s-nf (z) • ш«-^-n(z — Z). By Lemma 1 there is a resolvent Г0,..., Ts-n of the cycle Г such that
Гао,П.,аа-п = Ca »,...,«,-„ • S[n], S1n] = {|z1| = • • • = |zn| = 1}.
Since
j cc(z - Z) = (rs-n, - Z )> = 1,
from the Cauchy integral representation formula we get
J f (z)c(z - Z) = (rs-n, f (z) • - Z)> = f (Z).
□
References
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[2] R.Bott, L.W.Tu, Differential Forms in Algebraic Topology, Berlin, Springer-Verlag, 1982.
[3] V.M.Buchstaber, T.E.Panov, Torus Actions and Combinatorics of Polytopes, Proc. Steklov Inst. Math., 225(1999), 87-120.
[4] V.M.Buchstaber, T.E.Panov, Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series, vol. 24, American Mathematical Society, Providence, RI, 2002.
[5] D.A.Cox, The homogeneous coordinate ring of toric variety, J. Algebraic Geometry, 4(1995), 17-50.
[6] D.A.Cox, Recent developments in toric geometry, Algebraic geometry — Santa Cruz 1995, 389-436, Volume 2, AMS, Providence, RI, 1997, 389-436.
[7] A.M.Gleason, The Cauchy-Weil theorem, J. Math. Mech, 12(1963), 429-444.
[8] M.Goresky, R.MacPherson, Stratified Morse Theory, Berlin, Springer-Verlag, 1988.
[9] P.Griffiths, J.Harris, Principles of Algebraic Geometry, Wiley, 1994.
[10] A.Shchuplev, A.K.Tsikh, A.Yger, Residual kernels with singularities on coordinate planes, Proceedings of the Steklov Institute of Mathematics, 253(2006), 256-274.
[11] V.A.Vassiliev, Topology of plane arrangements and their complements, Russian Math. Surveys, 56(2001), no. 2, 365-401.
Фильтрация Ходжа на дополнениях к наборам комплексных координатных подпространств и интегральные представления голоморфных функций
Юрий В. Элияшев
В статье вычисляется фильтрация Ходжа на когомологиях дополнений к наборам комплексных координатных подпространств. Эти результаты используются для нахождения интегральных представлений голоморфных функций, в которых ядра имеют сингулярности на наборах координатных подпространств.
Ключевые слова: фильтрация Ходжа, конфигурации плоскостей, интегральные представления, торическая топология.
